Do you have to have a license to open a bed bug extermination business?

It's entirely dependent upon where you live. As this is the internet, you could live anywhere so it's difficult to advise via the internet.

If you are in Ontario, Canada... and wish to open a bed bug heat treatment business, without chemicals, then no. Otherwise, you need an employee with a structural exterminators license, and then you need to apply for the PCO (pest control operators) license.